Chapter 3.7, Problem 7E

Interpretation Introduction

Interpretation:

To sketch the graph of the nonlinear function $\text{g(p) = }\frac{{\text{βp}}^{\text{n}}}{{\text{K}}^{\text{n}}{\text{+p}}^{\text{n}}}$, for $\text{n >> 1}$ and to find the simple shape the function approaches as $\text{n}\to \infty$, if the basal expression is given as $\dot{\text{p}}\text{ = α +}\frac{{\text{βp}}^{\text{n}}}{{\text{K}}^{\text{n}}{\text{+p}}^{\text{n}}}\text{ - δp}$. The phase portrait for the system for $\text{δK - α > β}$, $\text{δK - α = }\frac{\text{β}}{2}$, and $\text{δK - α < 0}$ is to be plotted. To plot the bifurcation diagram for the system, assuming $\text{δK > β}$, and to indicate how the location and stability of the fixed points $\text{p*}$ vary with respect to $\text{α}$. To discuss how $\text{p}$ behaves if is very slowly

increased from $\text{α = 0}$ to $\text{α > δK}$, and then very slowlydecreased back to $\text{α = 0}$. To show that such a pulsed stimulation leads to hysteresis.

Concept Introduction:

The basal expression is

$\dot{\text{p}}\text{ = α +}\frac{{\text{βp}}^{\text{n}}}{{\text{K}}^{\text{n}}{\text{+p}}^{\text{n}}}\text{ - δp}$

Here $\text{α}$ is the basal transcription rate, $\text{β}$ is the maximal transcription rate, $\text{K}$ is the activation coefficient, and $\text{δ}$ is the decay rate of the protein.